All the goals came in the second half, with Jim O'Brien netting a double - ending a drought dating back to April - before Lukas Jutkiewicz added gloss to the scoreline.
The result got Motherwell's season back on track under caretaker manager Brown following five straight defeats and left new Killie boss Jimmy Calderwood in no doubt of the size of the task facing him at Rugby Park.
Both managers made a number of changes for their maiden team selections.
Calderwood recalled captain Kevin Kyle, Mark Burchill, Mehdi Taouil, Garry Hay, Simon Ford and Cameron Bell.
Brown - who was in the dugout for his side's last match at Hearts but not in control of team affairs - drafted in Keith Lasley, Giles Coke and John Sutton.
Saturday's match went ahead despite burst pipes forcing the closure of the stands behind both goals at Rugby Park.
The game was a meeting between the two most out-of-form clubs in the SPL, with second-bottom Killie having won only one of their previous nine games and Motherwell on an eight-match winless run.
Two of Brown's picks combined for the first effort on goal in the fourth minute when an under-pressure Sutton headed Lasley's cross over.
Jutkiewicz then scuffed a low shot wide from 18 yards before a vital Frazer Wright interception denied Sutton from point-blank range.
Jutkiewicz was inches from opening the scoring in the 14th minute with a left-foot strike from 10 yards as Motherwell continued to dominate.
Killie finally had a shot on goal six minutes later when Taouil curled just wide from the edge of the area.
And they could have led moments later when Craig Bryson was denied by John Ruddy, who made himself big enough to smother the midfielder's shot.
Opposite number Bell also needed to be sharp to turn the latest Jutkiewicz effort round his near post.
Ross Forbes then curled a free-kick inches wide, so close in fact the visiting fans briefly began celebrating.
Bell raced to the edge of his box to thwart Sutton before Killie began to work their way back into the game.
Kyle, who still faces disciplinary action from the club over his recent comments in the media, saw a header deflected just past the post.
Jamie Hamill's shot from range was too close to Ruddy but Motherwell had the final say before half-time when Tom Hateley's free-kick whistled just over the bar.
The second half saw both sides regularly surrender possession until 10 minutes in, when the visitors put together the best passing move of the match to take the lead.
The ball was beautifully worked forward to Forbes, who fed O'Brien to finish left-footed across Bell.
The midfielder was booked for going into the crowd to celebrate.
Burchill could easily have levelled moments later but he headed Taouil's cross over the bar.
Motherwell were then forced to replace Forbes with Chris Humphrey following a challenge by Hamill, who was soon booked for bringing down O'Brien.
Sutton glanced O'Brien's resulting free-kick just wide.
Kyle was yellow carded for dissent after appealing for a foul which was not given, before Calderwood threw on David Fernandez for Burchill 20 minutes in.
Manuel Pascali followed for James Fowler seconds after Kyle had headed over under pressure.
Fernandez almost made an instant impact when his finish from Taouil's cross was deflected just wide.
But the game looked over 20 minutes from time when Coke burst unchallenged down the Killie right before crossing for O'Brien to slide home his second.
Calderwood made his last change when Gavin Skelton came on for Invincibile, Brown responding by withdrawing O'Brien for Marc Fitzpatrick.
Jutkiewicz put the seal on an emphatic win for the visitors in the 78th minute when he found the corner of the net on the rebound after Sutton's initial shot had been blocked.
A mazy run and thumping angled drive from Humphrey almost made it 4-0 before Jutkiewicz came off for Robert McHugh.
Ruddy prevented Fernandez netting a late consolation, while Kyle's wayward last-minute effort summed up Killie's afternoon.